Reflection coefficients and screech-tone prediction in supersonic jets

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Reflection coefficients and screech-tone prediction in supersonic jets

Matteo Mancinelli, Vincent Jaunet, Peter Jordan, Aaron Towne, Steve Girard

To cite this version:

Matteo Mancinelli, Vincent Jaunet, Peter Jordan, Aaron Towne, Steve Girard. Reflection coefficients

and screech-tone prediction in supersonic jets. 25th AIAA/CEAS Aeroacoustics Conference, May

2019, Delft, Netherlands. �10.2514/6.2019-2522�. �hal-02378551�

Reflection coefficients and screech-tone prediction in supersonic jets

Matteo Mancinelli^∗

Institut Pprime, CNRS-Université de Poitiers-ENSMA, 86962 Chasseneuil du Poitou, Poitiers, France Direction des Lanceurs, CNES, 75612 Paris, France

Vincent Jaunet^†and Peter Jordan^‡

Institut Pprime, CNRS-Université de Poitiers-ENSMA, 86962 Chasseneuil du Poitou, Poitiers, France Aaron Towne^§

University of Michigan, Ann Arbor, MI 48109, USA Stève Girard^¶

Institut Pprime, CNRS-Université de Poitiers-ENSMA, 86962 Chasseneuil du Poitou, Poitiers, France

We study and model the resonance mechanism underlying the generation of A1 and A2 screech modes in an under-expanded supersonic jet. Following our previous work [1], where upstream-travelling guided jet modes were used to provide closure of the screech resonance loop, we here consider a more complete model in which both wavenumbers and frequencies of the upstream- and downstream-travelling waves are complex. The new model requires knowledge of the upstream and downstream reflection coefficients, which are treated as parameters and identified using the experimental data. The new screech model is shown to provide a more complete description of the measured data.

I. Nomenclature

c∞ = Ambient speed of sound

k = Dimensionless streamwise wavenumber ω = Non-dimensional frequency

m = Order of the azimuthal Fourier mode n_r = Radial order of the guided jet modes St = Nozzle diameter-based Strouhal number M_a = Acoustic Mach number

Mj = Jet Mach number Uj = Jet velocity

T = Jet-to-ambient temperature ratio k_{K H} = Kelvin-Helmholtz instability mode k_p = Guided jet mode

L₁ = Length of the first shock cell Ls = Location of thes^th shock cell

Re = Nozzle diameter-based Reynolds number D = Nozzle diameter

r = Radial distance from the nozzle axis SPSL = Sound Pressure Spectrum Level R₁R₂ = Reflection coefficient product

∗Post-doctoral Research Fellow, Département Fluides Thermique et Combustion, 11 Boulevard Marie et Pierre Curie.

†Assistant Professor, Département Fluides Thermique et Combustion, 11 Boulevard Marie et Pierre Curie.

‡Researcher, Département Fluides Thermique et Combustion, 11 Boulevard Marie et Pierre Curie.

§Assistant Professor, Department of Mechanical Engineering, 2350 Hayward Street.

¶Research Engineer, Département Fluides Thermique et Combustion, 11 Boulevard Marie et Pierre Curie.

II. Introduction

T

he generation and modelling of screech noise in non-perfectly expanded supersonic jets is a long-standing issue.

According to Powellet al. [2], the screech-frequency evolution with the jet Mach number is characterised by the jump into different modes or stages, namely axisymmetric A1 and A2 modes, flapping B and D modes and helical C mode.

It is known that screech is a resonance phenomenon due to a feedback mechanism loop involving a downstream- travelling flow disturbance generated at the nozzle exit and an upstream-travelling wave generated by the interaction of the flow disturbance with the shock-cell pattern. Since the seminal work of Powell [3], it has been assumed that the closure mechanism of the screech loop is provided by a free-stream acoustic wave. On the basis of this phenomenological description of the screech mechanism, several screech-frequency prediction models have been proposed [3–6]. The idea that screech may not be closed by free-stream acoustic waves was first suggested by Shen &

Tam [7], who claimed that A1 and B screech modes involved free-stream acoustic waves, whereas A2 and C modes involved upstream-travelling guided jet modes. These modes were first studied by Tam & Hu [8] and have been exploited to explain many resonance phenomena in jets (resonances occurring in subsonic and supersonic impinging jets [9, 10], weak resonances in high-speed subsonic jets [11], high-amplitude tone appearance in jet-flap interaction configuration [12]). Most importantly, they have been shown to be active in screeching supersonic jets [13, 14]. Given this observation, we recently developed a screech-frequency prediction model based on a resonance between downstream-travelling Kelvin-Helmholtz (K-H) wave and upstream-travelling guided jet modes [1]. Both K-H and upstream-travelling jet waves were computed via a cylindrical vortex-sheet model. Contrary to the assertions of Shen & Tam [7], we showed that both A1 and A2 screech modes are underpinned by a resonance involving the said upstream-travelling modes.

The predictions obtained provide better agreement with experimental data than does the classical screech-prediction approach using free-stream acoustic waves.

In the said model, and consistent with what is frequently assumed when considering fluid-mechanics resonance phenomena [12], we assumed that the upstream- and downstream-travelling waves are both spatially neutrally stable.

With these assumptions, knowledge of the magnitude of the reflection coefficients is not necessary for screech-frequency predictions. While predictions provide good agreement with data, the simplification misses certain aspects of the real behaviour of screech.

We thus here consider an improved screech-tone prediction model which brings us to focus on the upstream and downstream reflection mechanisms involved in screech. The model is based on a complex frequency-wavenumber analysis. The assumption of spatially neutrally stable waves is thus dropped. Both the magnitude and phase of the reflection coefficient product come into play in determination of the resonance frequency. The reflection coefficient product is an unknown function of both frequency and flow conditions. We therefore treat it as a parameter and we explore the prediction results obtained by imposing its amplitude and phase. On the basis of the results obtained, we propose a functional form of the reflection coefficient product as a function of both frequency and jet Mach number.

The paper is organised as follows. The resonance model is presented in §III. §IV describes the experimental setup and the instrumentation adopted. Main results concerning the screech-frequency predictions and the estimation of the reflection coefficient product are reported in §V. Conclusions and future developments of the research activity are discussed in §VI.

III. Resonance model

I

n this section we present the model based on a resonance between downstream-travelling K-H instability wave and upstream-travelling guided jet modes. Following Towneet al.[11] and Mancinelliet al.[1], we use the terms downstream- and upstream-travelling to designate the sign of the group velocity. Accordingly, the downstream- and upstream-travelling waves are indicated with superscripts+and−, respectively. The notation used in the manuscript is reported in the following for the sake of clarity. The Kelvin-Helmholtz instability is denotedk_{K H} and the guided jet modesk_p.

A. Cylindrical vortex-sheet model

T

he linear dynamics of the waves is modelled via a cylindrical vortex sheet [15, 16]. Following our previous work [1], the normal mode ansatz is:

q(x,r, θ,t)=qˆ(r)ei(k x+mθ−ωt), (1)

where k is the streamwise wavenumber normalised by the nozzle diameter D, mthe azimuthal mode andω the dimensionless frequencyω=2πSt M_a, withSt= f D/U_j the nozzle-diameter-based Strouhal number andM_a =U_j/c∞

the acoustic Mach number. The vortex-sheet dispersion relation is:

D(k, ω;M_a,T,m)= 1

1−^{k M}_ω^a2 +1 T

I_m ^γi

2

γo

2 K_m−1 ^γo

2

+mK_m ^γo

2

Km γo 2

γi 2Im−1

γi 2

−mIm γi 2

=0, (2)

with

γ_i = r

k²−1

T (ω−M_ak)², (3a)

γ_o =p

k²−ω², (3b)

where IandKare modified Bessel functions of the first and second kind, respectively, andT is the jet-to-ambient temperature ratioT_j/T∞, so that the relation between the acoustic and jet Mach numbers is given byM_j =U_j/c_j =M_a/√

T. The branch cut of the square root in eqs. (3a) and (3b) is chosen such that−π/2≤arg γ_i,o

≤π/2.

According to Towneet al. [11],k_pmodes belong to a hierarchical family of waves identified by their azimuthal and radial ordersmandn_r, respectively. We restrict attention to azimuthal modem=0 due to the axisymmetry property of screech modes A1 and A2, and we let vary the radial order in the rangen_r =1,2. We assume isothermal conditions in the modelling, i.e.T =1. Frequency/wavenumber pairs(ω,k)that satisfy eq. (2) define eigenmodes of the vortex sheet for given values ofm,M_aandT. To find these pairs, we specify a frequencyωand compute the associated eigenvalues kaccording to eq. (2).

In our previous work [1],ω ∈ R, consequentlyk⁺_{K H} ∈ Candk⁻_p ∈ R. Then a neutral mode assumption was made in order to compute resonance-frequency predictions neglecting the imaginary part of the K-H instability wave. In this paper we consider a complex frequencyω∈ C, this implies that both downstream- and upstream-travelling waves k⁺_{K H},k⁻_p ∈ C. In the complex-frequency analysis, eigenvalues must satisfy the dispersion relation (2) and at the same time the constraint imposed by the magnitude of the reflection coefficient product that is described in the next section.

B. Resonance conditions

I

n this section we discuss the conditions thatk_{K H}⁺ andk⁻_p must satisfy in order for resonance to occur. We assume that waves exchange energy upstream at the nozzle exit and downstream at thes^th shock-cell location where an interaction between the K-H instability wave and the shock-cell pattern occurs. According to the results we obtained in our previous work [1] and in agreement with results reported in the literature [17], we consider the fourth shock-cell as downstream reflection point, i.e.s=4. The location of thes^thshock cell is given by the Pack’s model [18]:

L₁ M_j = π 2.4048

q

M²_j −1, (4a)

L_s M_j =L₁((1−α)s+α), (4b) whereL₁is the length of the first shock cell,Lsthes^thshock-cell location andαthe rate of decrease of the shock-cell length with the downstream distance, whose value was set to 0.06 according to Harper-Bourne [19]. Following Landau

& Lifshitz [20], the condition to be satisfied for resonance to occur is:

R₁R₂e^{i∆k Ls} =1, (5)

whereR₁ St,Mj

∈ CandR₂ St,Mj,Ls

∈ Care the reflection coefficients at the nozzle exit and shock-cell location, respectively. Following Jordanet al.[12], eq. (5) can be rewritten in terms of magnitude and phase constraints,

e^∆kiLs =|R₁R₂|, (6a)

∆k_rL_s+φ=2pπ, (6b)

whereφis the phase of the reflection coefficient product.

Fig. 1 Vortex-sheet solutions for both k_{K H}⁺ and k⁻_p satisfying eq. (2) for azimuthal mode m = 0and fully expanded jet Mach number Mj =1.1. Upstream-travelling modes fornr =2are here considered. The colour map showsω_i, whereas the black contour representsSt=ω_r/2πM_a. Solid brown lines represent the eigenvalues in the case of real frequency analysis. Markers represent eigenvalues satisfying both the dispersion relation(2) and the magnitude constraint(6a)fors =4and different values of|R₁R₂|: magentarefer to|R₁R₂| =10⁻⁴, green◦to|R₁R₂|=10⁻³, blue4to|R₁R₂|=10⁻².

Perform resonance-frequency predictions involves finding triplets

k⁺_{K H},k⁻_p, ω

∈ Csimultaneously satisfying eqs.

(2), (6a) and (6b). This implies the knowledge of the reflection coefficient product as a function of both frequency and jet Mach number. Given that we do not have any knowledge aboutR₁R₂, we use the reflection coefficient product as a parameter of the model, we assign a value to it and we solve eqs. (2) and (6a) to find complex frequencies and eigenvalues constrained by the magnitude of the reflection coefficient product. Then we use the phase constraint in eq. (6b) to select resonance frequency. Following our previous work [1], we explore the extreme conditionsφ=0 andφ =−π. According to the results we found in Mancinelliet al. [1], we observed the best agreement with the experimental data by considering out-of-phase reflection conditions, which leads to the following resonance criterion,

Re

k⁺_{K H}−k⁻_p

=∆k_r= (2p+1)π

L_s . (7)

Figure 1 shows the vortex-sheet solutionsω(k)for both downstream- and upstream-travelling waves for azimuthal modem=0, fully expanded jet Mach numberM_j =1.1 and fourth shock-cell location as downstream reflection point.

Eigenvalues obtained from both real- and complex-frequency analyses are represented, withk_{K H}⁺ andk⁻_psatisfying both eqs. (2) and (6a) forω∈ C. We consider upstream-travelling modes of second radial order, i.e. nr =2. Eigenvalues are computed for three constant values of reflection coefficient product amplitude: |R₁R₂|=10⁻⁴, 10⁻³and 10⁻². We point out that both downstream- and upstream-travelling waves are characterised byk_i <0 forω∈ C, thus implying thatk⁺_{K H} andk⁻_p are spatially unstable and evanescent, respectively. We observe that forω∈ Cthe eigenvalues bend in thek_i-k_r plane towards regions of positive imaginary frequency with respect to the eigenvalues obtained forω ∈ R. Indeed, values ofω_i ≶0 represents attenuation/amplification in time of the resonance feedback loop. Eigenvalues satisfying both eqs. (2) and (6a) are therefore eigenvalues that, for the given value of|R₁R₂|, are characterised by positive values ofω_ito sustain resonance.

Figure 2 represents the resonance-frequency selection by imposing out-of-phase reflection conditions (7) form=0 andM_j =1.1. We consider the amplitude of the reflection coefficient product equal to 10⁻¹and 10⁻³fork⁻_p modes of the first and second radial orders, respectively.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0

2 4 6 8 10 12

Fig. 2 Resonance-frequency selection form = 0, M_j =1.1. Blue line refers to∆k_r for k⁻_p with n_r =1and

|R₁R₂| = 10⁻¹, red line to∆kr fork⁻_p withnr =2and |R₁R₂| = 10⁻³. Horizontal dashed black lines refer to resonance criteria in the case of out-of-phase reflection conditions(7).

IV. Experimental set-up

T

he experimental test campaign was performed at theSUCRÉ (SUpersoniC REsonance) jet noise facility in the Prométéetechnological platform of theInstitut Pprimein Poitiers. The supersonic under-expanded jet issues from a simple convergent nozzle of diameterD=0.01m. Experimental tests were carried out for a stagnation pressure range p₀=[1.89,5.75]with a corresponding fully expanded jet Mach number rangeM_j =[1,1.8]and a nozzle diameter-based Reynolds number rangeRe=U_jD/ν=

2.86·10⁵,7.9·10⁵

. The tests were performed with a very fine resolution

∆M_j =0.005 forM_j =[1,1.3]in order to capture the fine details of the Mach-number dependence of screech modes A1 and A2, and with a resolution∆M_j =0.01 forM_j =[1.31,1.8].

Pressure fluctuations were measured by GRAS 46BP microphones, whose frequency response is flat in the range 4H z-70k H z. Data were acquired by a National Instruments PXIe-1071 acquisition card with a sampling frequency of 200k H zwhich provides a maximum resolved Strouhal number range[2,3.2]well above theStof interest in this paper. The acquisition time was set equal to 30s, which is six orders of magnitude larger than the longest convective time, thus ensuring statistical convergence of the quantities presented in the paper [21]. An azimuthal array of six microphones was placed in the nozzle exit plane and radial distancer/D=1. Such a device allowed to resolve the most energetic azimuthal Fourier modes: m=0,±1,±2. A schematic representation of the experimental setup and microphone disposition is depicted in figure 3. For more details on the jet facility and experimental setup the reader can refer to Mancinelliet al.[1].

V. Results

F

igure 4 shows the Sound Pressure Spectrum Level [1, 22, 23] indB/Stas a function ofStandM_j for azimuthal modem=0. Signature of screech modes A1 and A2 are detected at high frequencies in the jet Mach number range [1.07,1.24]. A weaker low-frequency peak is detected as well for almost all the jet Mach numbers and its intensity rises with increasingM_j. Furthermore, we unexpectedly detect the footprint of the asymmetric B, C and D modes in the middle-frequency range. Indeed, the rising up of these asymmetric modes in the azimuthal modem=0 results from the non-linear interaction between azimuthal modesm= +1 andm=−1, as shown in the appendix. Hence, we did not take into account the signature of the B, C and D modes in the screech modelling ofm=0 mode and we focused on the predictions of screech modes A1 and A2 and on the low-frequency energy peak.

Figure 5 shows the spectral maps ofm=0 mode and the screech-frequency predictions obtained using the resonance model presented in §III. Predictions are obtained considering the fourth shock cell as downstream reflection point,

U

x r

microphone

Fig. 3 Schematic representation of the experimental setup and microphone disposition.

Fig. 4 Spectral map of modem=0and detection of screech modes of interest.

Fig. 5 Spectral maps of modem=0and screech-frequency predictions for different values of the amplitude of the reflection coefficient product. Predictions are represented with solid black lines. The region of allowable resonances for whichω_i ≥0is marked by dashed red lines.

out-of-phase reflection conditions in eq. (7) and upstream-travelling modes of radial ordern_r =1 and 2 for low-frequency and A1-A2 tones, respectively. The threshold frequenciesω_r|_ωi=0 for allM_j are superimposed on the plot as well.

These cut-off/cut-on frequencies defineSt-M_j region for whichω_i ≥0 and resonance is not damped. Predictions can be obtained only within this region, whose extension in theSt-M_j plane is a function of the amplitude of the reflection coefficient product. We explore four constant values of the reflection coefficient product amplitude, i.e. |R₁R₂|=10⁻⁴, 10⁻³, 10⁻²and 10⁻¹. We observe that for|R₁R₂|=10⁻⁴the lines individuatingω_r|_ωi=0do not include the signature of the A1 screech mode thus preventing the model to predict its resonance frequency. As the amplitude of the reflection coefficient product increases, the region of positive imaginary frequencies becomes wider incorporating allowable resonance frequencies for lower values ofMj. Specifically, the agreement of the resonance predictions of both A1 and A2 modes with the experimental data appears satisfactory for|R₁R₂|=10⁻³. For the highest value of the reflection coefficient product, i.e. |R₁R₂|=10⁻¹, we notice that the model provides unreliable predictions for the screech modes A1 and A2 and we observe the appearance of an allowable resonance-frequency region delimiting the low-frequency energy peak. Indeed, this peak is well-predicted in the rangeM_j =[1.4,1.65]. This result implies that|R₁R₂|=10⁻¹ has become too high to sustain resonances for the frequencies and jet flow conditions typical of A1 and A2 modes but suited for the low-frequency energy peak.

A. Predictions with frequency-dependent reflection coefficient product amplitude

T

he behaviour observed above suggests that the amplitude of the reflection coefficient product should have an increasing trend with the decreasing frequency. This assertion is in agreement with the behaviour observed in other physical systems, such as reflections in hard-walled cylindrical ducts (see among many Silvaet al. [24]), properties of sound-absorbing materials etc. Based on these evidences and on the results we found, we propose a function

|R₁R₂|=|R₁R₂(St) |having an increasing trend for decreasing frequency that fits the reflection coefficient product values which provided predictions at both high and low frequencies in figure 5. We opt for an exponential fit of the kind:

|R₁R₂(St) |=aexp(−b St), (8)

whereaandbare constants obtained by a least mean square optimization. The amplitude of the reflection coefficient product as a function of the frequency is reported in figure 6.

Fig. 6 Representation of the exponential fit based on the experimental results found in figure 5 proposed for the reflection coefficient product along the frequency.

Figure 7 shows the spectral map as a function ofStandM_j and the predictions obtained using the exponential shape of the reflection coefficient product along the frequency reported in eq. (8). We observe that the energies of both A1-A2 modes and low-frequency peak are bounded by theω_r|_ωi=0tracks, thus making both the spectral signatures predictable by the resonance model. Specifically, the low-frequency peak is well-predicted forM_j >1.4 and the predictions for A1 and A2 screech modes reproduce well the trend of the experimental data with a lower accuracy in the case of mode A2.

This result is ascribed to a not-perfectly suited value of the reflection coefficient product for the frequencies and jet flow conditions typical of mode A2. Indeed, this assertion is also confirmed by the fact that the positive-imaginary-frequency region, and so the region of allowable resonance frequencies, fork⁻_pmode of the second radial order extends forSt-M_j values where tones are not measured. It is here that the exigence of having a reflection coefficient product function of bothStandM_jbecomes apparent, as well as the fragility of the complex-frequency analysis due to the treatment of R₁R₂.

Figure 8 shows the contour map of ωi obtained using the frequency-dependent reflection coefficient product amplitude reported in eq. (8). Screech-frequency and low-frequency peak predictions are superimposed on the plot. We observe thatω_i has a larger value at high frequencies in correspondence of the screech tones than that found at low frequencies. The lower value ofω_i, which corresponds to a lower gain in time in the resonance feedback loop, most likely explains the lower energy amplitude of the low-frequency peak with respect to screech modes A1 and A2.

B. Reflection coefficient product evaluation

I

n order to try to find a better representation of the reflection coefficient product as a function of both frequency and jet flow conditions, we go back to the real-frequency analysis. We compute the eigenvalues of both the K-H and upstream-travelling modes forω∈ Rwithout making the neutral-mode assumption and so consideringk⁺_{K H} ∈ C and k⁻_p ∈ R. We select frequencies for which A1-A2 screech tones and low-frequency peak are measured, we compute the eigenvalues associated with these frequencies and we exploit the resonance condition in eq. (5), i.e.

R₁R₂e^{i∆k Ls} =1, to calculateR₁R₂∈ C. This calculation provides the reflection coefficient product forω_i =0, that is without amplification/attenuation in time of the resonance feedback loop, and hence provides the critical or minimum reflection coefficient product to sustain resonance at theStandM_j taken into account.

Figure 9 shows the amplitude of the computed critical reflection coefficient product as a function ofStandMjfor the frequencies where tones were measured experimentally. The outcome confirms more or less the order of magnitude of|R₁R₂|that we tested above in the frequency bands of interest. Specifically, we observe that|R₁R₂|increases with decreasing frequency as we assumed in §A and that|R₁R₂|exhibits a decreasing trend with increasingM_j. This result

Fig. 7 Spectral map of modem = 0and predictions obtained considering an exponential trend along the frequency of the amplitude of the reflection coefficient product.

Fig. 8 Contour map of the imaginary frequency for modem=0and screech-tone predictions obtained using an exponential fit of the reflection coefficient product amplitude along the frequency.

Fig. 9 Representation of the critical or minimum amplitude of the reflection coefficient product computed from eq.(5)consideringω∈ R,k⁺_{K H} ∈ Candk⁻_p ∈ Rfor azimuthal modem=0.

suggests a functional form of|R₁R₂ Mj

|to be used in order to define a more accurate reflection coefficient product map as a function of both frequency and flow conditions.

C. Refined screech-frequency predictions

O

n the basis of the outcome reported in figure 9, we propose here a semi-empirical shape of the amplitude of the reflection coefficient product as a function of both Strouhal and jet Mach numbers. The proposed functional form is:

|R₁R₂ St,M_j

|=aexp

−b M_j

·St^c(Mj)

(9) wherea,b Mj

andc Mj

are parameters determined via a least-mean-square-based fit of the critical|R₁R₂|values in figure 9. We impose a linear trend along the jet Mach number of the coefficientsb M_j

andc M_j

. The amplitude of the reflection coefficient product obtained is reported in figure 10. The spectral map and the predictions obtained using

|R₁R₂|=|R₁R₂ St,M_j

|are reported in figure 11. Lines indicatingω_r|_ωi=0are superimposed on the plot as well. We observe that the regions of the allowable resonance frequencies for bothn_r =1,2 are more tightly-fitted to the measured screech and low-frequency peaks, although a region ofω_i ≥0 still remains forn_r =2 and highM_jwhere no tones are measured. Furthermore, the accuracy of the resonance-frequency predictions is higher especially for the screech modes A1 and A2. This result underlines the potentialities of the linear model we propose to correctly describe and model the resonance dynamics of the axisymmetric screech modes.

VI. Conclusions

T

he resonance tones of the axisymmetric azimuthal mode of a supersonic jet have been modelled. Attention was focused on the A1-A2 screech modes and the low-frequency energy peak measured in the near pressure field of an under-expanded single supersonic jet. The work is a continuation of our recent study [1] in which we proposed a novel screech-frequency prediction model in which closure of the screech resonance loop is provided by upstream-travelling guided jet modes. In this paper we propose a more complete model in which both wavenumber and frequency are complex quantities. The model can provide a more complete description of what is observed experimentally, but to do so it requires knowledge of the upstream and downstream reflection coefficients, which appear in the model as the productR₁R₂. We treat this as a parameter. Results suggest the frequency and Mach number dependence of the reflection-coefficient product and a functional form is proposed that provides good agreement with measurements.

Fig. 10 Representation of the semi-empirical shape of the reflection coefficient product amplitude as a function of bothStandMj based on the trend of the critical|R₁R₂|reported in figure 9.

Fig. 11 Spectral map of azimuthal mode m = 0and screech-tone predictions obtained using a reflection coefficient product amplitude function of both Strouhal and jet Mach numbers. Solid black lines represent screech-frequency and low-frequency peak predictions, dashed red lines delimit theSt-M_jregion whereω_i ≥0.

Fig. 12 Cross-bicoherence between azimuthal modesm=1,m=−1andm=0for jet Mach numberM_j =1.3.

Appendix

T

he rising up of the footprint of the asymmetric B, C and D modes in the spectral content of modem=0 due to non-linear interaction between modesm=1 andm=−1 was proved computing the cross-bicoherence between these azimuthal modes. Non-linear quadratic interaction between azimuthal modes occurs whenm₁+m−1−m₀ =0.

Accordingly, the cross-bispectrum is computed as follows [25],

B_c(f,m₁,m−1,m₀)=hpˆ₁(f)pˆ−1(f)pˆ^∗₀(f)i (10) where ˆp_m(f)is the Fourier transform of them^thazimuthal mode and the superscript∗indicates complex conjugate.

The cross-bicoherence is obtained normalising the cross-bispectrum as follows, b²_c(f,m₁,m−1,m₀)= |B_c(f,m₁,m−1,m₀) |²

h|pˆ₀(f) |²ih|pˆ₁(f)pˆ−1(f) |²i (11) Figure 12 shows the cross-bicoherence as a function of Strouhal number between azimuthal modesm=1,m=−1 andm=0 forMj =1.3. For this jet Mach number only screech mode B is active and its signature should be found in m=1 andm=−1 modes. We observe that a unitary bicoherence level is found forSt₁=St₂ ≈0.34. This implies that a strong non-linear interaction occurs between modesm=1 andm=−1 at the frequency componentSt≈0.34 leading to the appearance of an energy signature in modem=0 for such frequency content, as confirmed by figure 4. We also observe a lower bicoherence level for frequency componentsSt₁≈0.68 andSt₂≈0.34 related to the interaction of the B screech mode with its first harmonic. The signature of this interaction is detected in the spectral map in figure 4 by the appearance of the first harmonic of B screech mode in the energy content of azimuthal modem=0.

Acknowledgments

M.M. acknowledges the support of Centre National d’Études Spatiales (CNES) under a post-doctoral grant.

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